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## G = D20⋊22D4order 320 = 26·5

### 10th semidirect product of D20 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊22D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — D20⋊22D4
 Lower central C5 — C2×C10 — D20⋊22D4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for D2022D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a9, cbc-1=a8b, dbd=a18b, dcd=c-1 >

Subgroups: 1078 in 292 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×14], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×21], D4 [×14], Q8 [×4], C23, C23 [×3], D5 [×5], C10 [×3], C10, C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4, C22×C4 [×7], C2×D4 [×6], C2×Q8, C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×5], D10 [×4], D10 [×7], C2×C10, C2×C10 [×3], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic10 [×2], C4×D5 [×14], D20 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22×C10, D46D4, C4×Dic5, C10.D4, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×4], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5 [×6], C2×D20, C2×D20 [×2], C4○D20 [×4], Q82D5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, Q8×C10, D10.12D4 [×2], D10⋊D4 [×2], D5×C4⋊C4 [×2], D208C4, D10.13D4 [×2], D102Q8, C4×C5⋊D4, Dic5⋊Q8, C5×C22⋊Q8, C2×C4○D20, C2×Q82D5, D2022D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, C22×D5 [×7], D46D4, D4×D5 [×2], C23×D5, C2×D4×D5, Q8.10D10, D5×C4○D4, D2022D4

Smallest permutation representation of D2022D4
On 160 points
Generators in S160
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 123)(2 122)(3 121)(4 140)(5 139)(6 138)(7 137)(8 136)(9 135)(10 134)(11 133)(12 132)(13 131)(14 130)(15 129)(16 128)(17 127)(18 126)(19 125)(20 124)(21 65)(22 64)(23 63)(24 62)(25 61)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 73)(34 72)(35 71)(36 70)(37 69)(38 68)(39 67)(40 66)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)(101 146)(102 145)(103 144)(104 143)(105 142)(106 141)(107 160)(108 159)(109 158)(110 157)(111 156)(112 155)(113 154)(114 153)(115 152)(116 151)(117 150)(118 149)(119 148)(120 147)
(1 107 69 95)(2 116 70 84)(3 105 71 93)(4 114 72 82)(5 103 73 91)(6 112 74 100)(7 101 75 89)(8 110 76 98)(9 119 77 87)(10 108 78 96)(11 117 79 85)(12 106 80 94)(13 115 61 83)(14 104 62 92)(15 113 63 81)(16 102 64 90)(17 111 65 99)(18 120 66 88)(19 109 67 97)(20 118 68 86)(21 48 127 148)(22 57 128 157)(23 46 129 146)(24 55 130 155)(25 44 131 144)(26 53 132 153)(27 42 133 142)(28 51 134 151)(29 60 135 160)(30 49 136 149)(31 58 137 158)(32 47 138 147)(33 56 139 156)(34 45 140 145)(35 54 121 154)(36 43 122 143)(37 52 123 152)(38 41 124 141)(39 50 125 150)(40 59 126 159)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 35)(22 24)(23 33)(25 31)(26 40)(27 29)(28 38)(30 36)(32 34)(37 39)(41 151)(42 160)(43 149)(44 158)(45 147)(46 156)(47 145)(48 154)(49 143)(50 152)(51 141)(52 150)(53 159)(54 148)(55 157)(56 146)(57 155)(58 144)(59 153)(60 142)(61 77)(62 66)(63 75)(65 73)(67 71)(68 80)(70 78)(72 76)(81 101)(82 110)(83 119)(84 108)(85 117)(86 106)(87 115)(88 104)(89 113)(90 102)(91 111)(92 120)(93 109)(94 118)(95 107)(96 116)(97 105)(98 114)(99 103)(100 112)(121 127)(122 136)(123 125)(124 134)(126 132)(128 130)(129 139)(131 137)(133 135)(138 140)```

`G:=sub<Sym(160)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,123)(2,122)(3,121)(4,140)(5,139)(6,138)(7,137)(8,136)(9,135)(10,134)(11,133)(12,132)(13,131)(14,130)(15,129)(16,128)(17,127)(18,126)(19,125)(20,124)(21,65)(22,64)(23,63)(24,62)(25,61)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147), (1,107,69,95)(2,116,70,84)(3,105,71,93)(4,114,72,82)(5,103,73,91)(6,112,74,100)(7,101,75,89)(8,110,76,98)(9,119,77,87)(10,108,78,96)(11,117,79,85)(12,106,80,94)(13,115,61,83)(14,104,62,92)(15,113,63,81)(16,102,64,90)(17,111,65,99)(18,120,66,88)(19,109,67,97)(20,118,68,86)(21,48,127,148)(22,57,128,157)(23,46,129,146)(24,55,130,155)(25,44,131,144)(26,53,132,153)(27,42,133,142)(28,51,134,151)(29,60,135,160)(30,49,136,149)(31,58,137,158)(32,47,138,147)(33,56,139,156)(34,45,140,145)(35,54,121,154)(36,43,122,143)(37,52,123,152)(38,41,124,141)(39,50,125,150)(40,59,126,159), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,35)(22,24)(23,33)(25,31)(26,40)(27,29)(28,38)(30,36)(32,34)(37,39)(41,151)(42,160)(43,149)(44,158)(45,147)(46,156)(47,145)(48,154)(49,143)(50,152)(51,141)(52,150)(53,159)(54,148)(55,157)(56,146)(57,155)(58,144)(59,153)(60,142)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)(81,101)(82,110)(83,119)(84,108)(85,117)(86,106)(87,115)(88,104)(89,113)(90,102)(91,111)(92,120)(93,109)(94,118)(95,107)(96,116)(97,105)(98,114)(99,103)(100,112)(121,127)(122,136)(123,125)(124,134)(126,132)(128,130)(129,139)(131,137)(133,135)(138,140)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,123)(2,122)(3,121)(4,140)(5,139)(6,138)(7,137)(8,136)(9,135)(10,134)(11,133)(12,132)(13,131)(14,130)(15,129)(16,128)(17,127)(18,126)(19,125)(20,124)(21,65)(22,64)(23,63)(24,62)(25,61)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147), (1,107,69,95)(2,116,70,84)(3,105,71,93)(4,114,72,82)(5,103,73,91)(6,112,74,100)(7,101,75,89)(8,110,76,98)(9,119,77,87)(10,108,78,96)(11,117,79,85)(12,106,80,94)(13,115,61,83)(14,104,62,92)(15,113,63,81)(16,102,64,90)(17,111,65,99)(18,120,66,88)(19,109,67,97)(20,118,68,86)(21,48,127,148)(22,57,128,157)(23,46,129,146)(24,55,130,155)(25,44,131,144)(26,53,132,153)(27,42,133,142)(28,51,134,151)(29,60,135,160)(30,49,136,149)(31,58,137,158)(32,47,138,147)(33,56,139,156)(34,45,140,145)(35,54,121,154)(36,43,122,143)(37,52,123,152)(38,41,124,141)(39,50,125,150)(40,59,126,159), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,35)(22,24)(23,33)(25,31)(26,40)(27,29)(28,38)(30,36)(32,34)(37,39)(41,151)(42,160)(43,149)(44,158)(45,147)(46,156)(47,145)(48,154)(49,143)(50,152)(51,141)(52,150)(53,159)(54,148)(55,157)(56,146)(57,155)(58,144)(59,153)(60,142)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)(81,101)(82,110)(83,119)(84,108)(85,117)(86,106)(87,115)(88,104)(89,113)(90,102)(91,111)(92,120)(93,109)(94,118)(95,107)(96,116)(97,105)(98,114)(99,103)(100,112)(121,127)(122,136)(123,125)(124,134)(126,132)(128,130)(129,139)(131,137)(133,135)(138,140) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,123),(2,122),(3,121),(4,140),(5,139),(6,138),(7,137),(8,136),(9,135),(10,134),(11,133),(12,132),(13,131),(14,130),(15,129),(16,128),(17,127),(18,126),(19,125),(20,124),(21,65),(22,64),(23,63),(24,62),(25,61),(26,80),(27,79),(28,78),(29,77),(30,76),(31,75),(32,74),(33,73),(34,72),(35,71),(36,70),(37,69),(38,68),(39,67),(40,66),(41,94),(42,93),(43,92),(44,91),(45,90),(46,89),(47,88),(48,87),(49,86),(50,85),(51,84),(52,83),(53,82),(54,81),(55,100),(56,99),(57,98),(58,97),(59,96),(60,95),(101,146),(102,145),(103,144),(104,143),(105,142),(106,141),(107,160),(108,159),(109,158),(110,157),(111,156),(112,155),(113,154),(114,153),(115,152),(116,151),(117,150),(118,149),(119,148),(120,147)], [(1,107,69,95),(2,116,70,84),(3,105,71,93),(4,114,72,82),(5,103,73,91),(6,112,74,100),(7,101,75,89),(8,110,76,98),(9,119,77,87),(10,108,78,96),(11,117,79,85),(12,106,80,94),(13,115,61,83),(14,104,62,92),(15,113,63,81),(16,102,64,90),(17,111,65,99),(18,120,66,88),(19,109,67,97),(20,118,68,86),(21,48,127,148),(22,57,128,157),(23,46,129,146),(24,55,130,155),(25,44,131,144),(26,53,132,153),(27,42,133,142),(28,51,134,151),(29,60,135,160),(30,49,136,149),(31,58,137,158),(32,47,138,147),(33,56,139,156),(34,45,140,145),(35,54,121,154),(36,43,122,143),(37,52,123,152),(38,41,124,141),(39,50,125,150),(40,59,126,159)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,35),(22,24),(23,33),(25,31),(26,40),(27,29),(28,38),(30,36),(32,34),(37,39),(41,151),(42,160),(43,149),(44,158),(45,147),(46,156),(47,145),(48,154),(49,143),(50,152),(51,141),(52,150),(53,159),(54,148),(55,157),(56,146),(57,155),(58,144),(59,153),(60,142),(61,77),(62,66),(63,75),(65,73),(67,71),(68,80),(70,78),(72,76),(81,101),(82,110),(83,119),(84,108),(85,117),(86,106),(87,115),(88,104),(89,113),(90,102),(91,111),(92,120),(93,109),(94,118),(95,107),(96,116),(97,105),(98,114),(99,103),(100,112),(121,127),(122,136),(123,125),(124,134),(126,132),(128,130),(129,139),(131,137),(133,135),(138,140)])`

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 10 10 10 10 20 2 2 2 2 4 4 4 4 10 10 10 10 20 20 20 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D10 2- 1+4 D4×D5 Q8.10D10 D5×C4○D4 kernel D20⋊22D4 D10.12D4 D10⋊D4 D5×C4⋊C4 D20⋊8C4 D10.13D4 D10⋊2Q8 C4×C5⋊D4 Dic5⋊Q8 C5×C22⋊Q8 C2×C4○D20 C2×Q8⋊2D5 D20 C22⋊Q8 Dic5 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C4 C2 C2 # reps 1 2 2 2 1 2 1 1 1 1 1 1 4 2 4 4 6 2 2 1 4 4 4

Matrix representation of D2022D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 37 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 0 0 0 40 6
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 31 6 0 0 0 0 4 10 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 37 0 0 0 0 26 29 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 6 40
,
 40 0 0 0 0 0 35 1 0 0 0 0 0 0 1 5 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 6 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,37,9,0,0,0,0,0,0,0,40,0,0,0,0,1,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,4,0,0,0,0,6,10,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,26,0,0,0,0,37,29,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,5,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;`

D2022D4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_{22}D_4`
`% in TeX`

`G:=Group("D20:22D4");`
`// GroupNames label`

`G:=SmallGroup(320,1303);`
`// by ID`

`G=gap.SmallGroup(320,1303);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,570,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^9,c*b*c^-1=a^8*b,d*b*d=a^18*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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